# Maths - Bicrossed Product

### Direct Product

Perhaps the simplest way to combine (multiply) two sets (and therefore groups) is to use the Cartesian product as explained on this page. This product produces a set from the product:

g×h = {g, h}

So the result of this product is a different type of entity than the elements being multiplied, so the multiplication is not closed, and therefore does not represent a group. The external product makes this into a group because the inputs to the multiplication are also sets:

{y,z}{y',z' } = {y(z • y'),zy' z'}

where:

• × is the operation of the combined algebra.
• * is the operation of the group G.
• o is the operation of the group H which may be, or may not be, the same as *.

### Example C2×C3

In order to try to understand this product of two groups lets try multiplying two very simple groups together, the simplest groups I can think of are C2 and C3.

#### C2

generator cayley graph table permutation representation
<m | m²>
 1 m m 1
< ( 1 2 ) >
 0 1 1 0

#### C3

generator cayley graph table permutation representation
<r | r³>
 1 r r² r r² 1 r² 1 r
< ( 1 2 3 ) >
 0 0 1 1 0 0 0 1 0

#### direct product C3 × C2

This gives :

generator cayley graph table
<m,r | m²,r³,rm=mr>
 {1,1} {r,1} {r²,1} {1,m} {r,m} {r²,m} {r,1} {r²,1} {1,1} {r,m} {r²,m} {1,m} {r²,1} {1,1} {r,1} {r²,m} {1,m} {r,m} {1,m} {r,m} {r²,m} {1,1} {r,1} {r²,1} {r,m} {r²,m} {1,m} {r,1} {r²,1} {1,1} {r²,m} {1,m} {r,m} {r²,1} {1,1} {r,1}
permutation representation
<(1 2 3)(4 5 6),(1 4)(2 5)(3 6)>
[
 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0
,
 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0
]

Note that, in addition to applying both the generators and constrains for the original groups we have had to apply an additional constraint: rm=mr. If we had not done this we would have the infinite free product.

## Generating a Bicrossed Product using a Program

We can use a computer program to generate these groups, here I have used Axiom/FriCAS which is described here.

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